Let's solve **Question 10 step by step**: ### Step 1: Understand the Problem - **Converter type:** Buck-boost - **Input voltage (\(V_{in}\))**: 12 V - **Output voltage (\(V_{out}\))**: -12 V - **Output current (\(I_{out}\))**: 20 A (magnitude) - **Switching frequency (\(f_s\))**: 100 kHz - **tan δ (dissipation factor) at 100 kHz**: 0.10 for available capacitors - **Required:** What value of output capacitor \(C\) is needed? Use a reasonable ripple specification. --- ### Step 2: Output Voltage Ripple Specification A typical output voltage ripple (\(\Delta V_{out}\)) is 1% to 2% of output voltage. Let's use **1%** as a reasonable ripple specification: \[ \Delta V_{out} = 0.01 \times |V_{out}| = 0.01 \times 12V = 0.12V \] **Explanation:** The ripple voltage is typically kept at 1% of the output voltage for good performance. --- ### Step 3: Buck-Boost Capacitor Ripple Calculation For a buck-boost converter, the output capacitor must supply current during the switching cycle, leading to ripple. The **peak-to-peak output voltage ripple** due to the capacitor is: \[ \Delta V_{out} = \frac{D \cdot I_{out}}{f_s \cdot C} \] where - \( D \) = Duty cycle - \( I_{out} \) = Output current - \( f_s \) = Switching frequency - \( C \) = Capacitance **Explanation:** This is the standard ripple equation for the output capacitor in a buck-boost converter. --- ### Step 4: Find Duty Cycle (\(D\)) For a buck-boost converter: \[ V_{out} = -\frac{D}{1 - D} V_{in} \] \[ -12 = -\frac{D}{1 - D} \cdot 12 \] \[ 1 = \frac{D}{1 - D} \] \[ 1 - D = D \] \[ 1 = 2D \implies D = 0.5 \] **Explanation:** We solve for the duty cycle using the buck-boost voltage relationship. --- ### Step 5: Substitute Values \[ \Delta V_{out} = \frac{D \cdot I_{out}}{f_s \cdot C} \] \[ 0.12 = \frac{0.5 \times 20}{100,000 \times C} \] \[ 0.12 = \frac{10}{100,000 \times C} \] \[ 0.12 \times 100,000 \times C = 10 \] \[ 12,000 \times C = 10 \] \[ C = \frac{10}{12,000} \] \[ C = 0.000833~\text{F} = 833~\mu\text{F} \] **Explanation:** Plugging values into the ripple formula gives the minimum capacitance needed for the given ripple. --- ### Step 6: Check ESR Losses (tan δ) The **equivalent series resistance (ESR)** of the capacitor causes additional ripple: \[ \text{ESR} = \frac{\tan \delta}{2\pi f_s C} \] Calculate ESR: \[ \text{ESR} = \frac{0.10}{2\pi \times 100,000 \times 833 \times 10^{-6}} \] \[ = \frac{0.10}{2\pi \times 100,000 \times 0.000833} \] \[ = \frac{0.10}{523.6} \] \[ = 0.000191~\Omega = 0.191~m\Omega \] Ripple due to ESR: \[ \Delta V_{ESR} = I_{out} \times ESR = 20 \times 0.000191 = 0.00382~V \] This is much less than our ripple spec (0.12 V), so it's **acceptable**. --- ### **Final Answer** **The required output capacitor value is approximately \( \boxed{833~\mu\text{F}} \) (or higher), using a capacitor with \(\tan\delta = 0.10\) at 100 kHz.** - This will keep the output voltage ripple within 1% of the output voltage. - The ESR at this tan δ and capacitance is very small and does not significantly contribute to the total ripple.

**Final Answer for Question 10:** The required output capacitor value is approximately **833 μF**. This value ensures the output voltage ripple remains within the specified 1% (about 0.12 V) for a buck-boost converter supplying 20 A at 12 V with a switching frequency of 100 kHz. Given the capacitor's dissipation factor (\(\tan \delta = 0.10\)), the ESR-related ripple is negligible, confirming that a capacitor of around 833 μF or higher is suitable. --- **Note:** Always select a capacitor with a voltage rating exceeding the output voltage (i.e., at least 25 V for safety margin) and consider real-world tolerances and temperature effects.

The solution employs the fundamental principles of switching power supply design, particularly the ripple calculation in output capacitors of a buck-boost converter. It starts by estimating a reasonable voltage ripple (1% of output voltage) and uses the standard capacitor ripple formula, which relates ripple voltage to the output current, switching frequency, and capacitance. The duty cycle is determined from the buck-boost relationship between input and output voltages, confirming D=0.5 for the given voltages. Substituting known values into the ripple equation yields the required capacitance, approximately 833 μF, ensuring the ripple stays within the specified limit. The analysis also examines the ESR contribution based on the given dissipation factor (\(\tan \delta\)), confirming its negligible effect at this capacitance. This approach combines basic converter theory, ripple specifications, and capacitor parameters to determine the appropriate output capacitance.